Method for evaluating and monitoring formation fracture treatment closure rates and pressures using fluid pressure waves

ABSTRACT

A method for characterizing a hydraulic fracture in a subsurface formation, includes inducing a pressure change in a well drilled through the subsurface formation. Pressure and/or a time derivative thereof is measured at a location proximate to a wellhead for a selected length of time. A conductivity of at least one fracture is determined using the measured at least one of pressure and the time derivative of pressure. A change in the determined conductivity with respect to time is determined.

CROSS REFERENCE TO RELATED APPLICATIONS

Continuation of International Application No. PCT/US2017/047488 filed onAug. 18, 2017. Priority is claimed from U.S. Provisional Application No.62/376465 filed on Aug. 18, 2016 and from International Application No.PCT/US2017/031507 filed on May 8, 2017. All of the foregoingapplications are incorporated herein by reference in their entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable

NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT

Not Applicable.

BACKGROUND

This disclosure relates to the field of borehole acoustic analysis andhydraulic fractures as well as hydraulic fracturing process monitoringand evaluation. In particular, the monitoring can be in real time whilehydraulic stimulation takes place, while additional analysis of the dataor comparisons with prior models can also be performed at another time.

This disclosure also relates to the field of seismic analysis ofhydraulic fractures. More specifically, the disclosure relates to methodfor analyzing geophysical properties of hydraulic fracture by analysisof acoustic pressure wave reflection and resonance.

Furthermore, this disclosure also relates to measurements of fracture(network) connectivity to wellbore and fracture (network) connectivityto the external reservoir volume.

Hydraulic fracturing has recently accounted for a significant growth ofunconventional (tight, shale) reservoir production in the United States.During hydraulic fracturing, fluid under high pressure is pumped into alow permeability reservoir to initiate fractures that tend to propagatebased on dominant stress geometries and stress distribution in thereservoir. To maintain connectivity and potential fluid (reservoirhydrocarbons and trapped fluids) flow through the fractures created bythe fluid under pressure, proppant is carried with the fracturing fluid.Proppant includes specific-sized sand or engineered (e.g. to withstandvery high pressure) compounds such as ceramics, coated sands, andothers. The proppant is injected along with the fracturing fluid(typically water and some chemicals that may include friction reducers,viscosifiers, gels, acid to help dissolve rock, etc.). Even thoughsimulations and rock physics/fracture propagation models have shed somelight on fracture creation and growth, many parameters of and forsuccessful/productive fracturing in terms of ultimate hydrocarbonproduction and recovery have typically been determined experimentallyand often by trial and error.

There are several ways known to create fracture networks in “stages” orsections moving from toe to heel (deepest point and the beginning of thehorizontal section of a highly inclined or horizontal well), typicallyreferred to as “plug and perf” and sliding sleeve (or similar) methods,that open only a small portion or section of the well or of perforations(openings) to the formation. Methods according to the present disclosureare applicable to plug and perf as well as sliding sleeve methodsbecause measurements can take place before, during and after the pumpingof fracturing fluid irrespective of the specific treatment or pumpingmethod used in a given section of a well.

Despite recent improvements in understanding production fromunconventional fractured reservoirs, current monitoring methods andanalysis, such as the passive or “microseismic” monitoring have beenless than optimal in obtaining efficient fluid recovery. It has beenestimated that only a fraction of stages in a multiple stage fracturedwell contribute significantly to ultimate hydrocarbon production.Moreover, fracture connectivity (related to permeability) and nearwell-bore fracture complexity (affecting efficient drainage) seem toshow impact on ultimate recovery but are difficult to both infer andmeasure and thus to design for with currently available methods.

The problem of efficient monitoring to optimize fracture treatmentdesign has been approached in many different ways using microseismic andother forms of monitoring (electromagnetic, downhole measurements andlogs, or, for example analyses using conductive or activated proppants).Such methods provide some level of information and detail, but haveseveral drawbacks. Typical microseismic or electromagnetic monitoringmethods require many sensors, significant processing time and computingresources, and can be labor intensive. In general, such methods can addsubstantial cost, time and labor to the process. In particular,additional significant post-acquisition processing of acquired data toobtain results makes real-time information availability limited orimpracticable.

U.S. Patent Application Publication No. 2013/0079935 A1 by Kabannik etal. describes a method using geophones and locates sensors inside awellbore. The disclosed method does not require any downhole sensors,even though such implementation may enhance some results and is notlimited to. Any downhole sensors are operationally difficult to deployand increase costs of measurements. Moreover, the method disclosed inthe '935 publication relies on more complex models and requiredinterrupting fracture pumping operations. Furthermore, the first part ofthe disclosed method is not concerned with determining the location ofmicroseismic events, only their detection.

A method for hydraulic impedance testing disclosed in Holzhausen, U.S.Pat. No. 4,802,144, relates to a method for analysis of freeoscillations of a connected well-fracture system, the latter of which isassumed to support wave propagation, to obtain fracture geometry (suchas length, height and width) by matching the data to pre-existing modelsor by inversion for the fracture geometry. The '144 patent does notdescribe either the effects of fracture permeability, nor inversion forwellbore-only parameters, such as tube wave velocity and attenuation.

With reference to U.S. Patent Application Publication No. 2011/0272147A1, by Beasley et al., the focus of such publication is on sensorsdisposed near a reservoir but not necessarily sensors hydraulicallyconnected to the reservoir Beasley et al. discloses performingmeasurement before and post hydraulic fracturing/stimulation operation.Moreover, the method disclosed in the '147 publication may not besuitable for rapid interpretation.

U.S. Patent Application Publication No. 2012/0069707 discloses usingmultiple receivers that are ground based, not connected hydraulically tothe wellbore, while also requiring reference data and models.

U.S. Patent Application Publication No. 2014/0216729 by McKenna focuseson determining a fracture network volume using microseismic eventtriangulation and detection from surface based ground sensors, ratherthan from a direct fluid connectivity of wellbore fluid with thefracture network as the present invention.

U.S. Pat. Nos. 4,907,204 and 7,035,165 B2 are both based on activeseismic well sources and well logging inside a wellbore, which useswireline or similar devices to traverse a borehole and as such may besignificantly more expensive and complex to implement in comparison witha single (or only a few) surface based borehole sensor(s). Additionalrelated art includes U.S. Pat. No. 7,819,188 B2.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example embodiment of a data acquisition system that maybe used in accordance with the present disclosure.

FIG. 2 shows an example geophysical model of subsurface formations beingfractured and measurements made according to the disclosure tocharacterize the fractures. It also shows the resonances driven infractures through pumping and microseismic activity.

FIG. 3 shows an example of data recording and analysis. The top frameshows pressure at a selected position in or along a well (arbitraryunits), the middle frame shows hydrophone or acoustic pressure change(time derivative) data, the bottom frame shows examples ofcharacteristic times and events.

FIG. 4A shows a graph of a representative active source hydrophone timeseries.

FIG. 4B illustrates how conductivity kw affects waveforms.

FIG. 4C shows how the reflection coefficient R depends on kw.

FIG. 5A shows the hydrophone Fourier spectrum from water hammer.

FIG. 5B shows the sensitivity of modeled spectra to fractureconductivity kw.

FIG. 6 shows several post shut-in fracture conductivity measurements ofa fracture treatment stage. Note the decrease corresponding to fractureclosures. Other formation effects affecting conductivity may be measuredover time post shut-in. Measurements of conductivity performed using thedescribed method.

FIG. 7 shows two example fracture conductivities plotted graphicallyover time.

Comparison of gel breakdown time between two stages on a same well isillustrated. Gel breakdown occurs where the dramatic increase inconductivity happens. Both example stages had gel pumped in.Measurements of conductivity performed using the described method.

FIG. 8 shows an example computer system that may be used in someembodiments.

DETAILED DESCRIPTION

The discussion below uses specific examples but is not necessarily theonly intended or possible implementation or use of the disclosedmethods. A person having skill in the art can devise similarimplementations to the same goals. Methods according to this disclosuremake practical use of pressure waves and pressure disturbances infracture(s), including the resonance of the combined well-fracturenetwork system, to determine hydraulic fracture network parameters.

During hydraulic fracturing, formations crack or fracture, and fluidwith proppant is injected in the opened cracks or fractures. Becausefractures may create an interconnected network, the terms “fracture” and“fracture network” may be used synonymously in the description below.Note that given the quantity of injected fluids, there is ageostatistical component and superposition to the sum of fracture sizesand distribution. Also note that this method is applicable to vertical,horizontal, or any other deviated well that undergoes hydraulicfracturing (stimulation) treatment.

Active sources can be water hammer, fracture treatment pumps, pistons,or other type sources specifically designed to generate tube waves,Stoneley waves, or borehole resonances, etc. as described herein below.

Continuous/passive sources are embedded in the operation itself and mayinclude general pumping noise, microseismic events, other geologicalphenomena not generally related to the fracturing operation (e.g.natural seismicity).

Fractures created during hydraulic fracture fluid pumping may beconnected to the wellbore through casing perforations, or slotted-sleeveports integrated into the completion casing and, if existing, anypreviously created or naturally existing fracture network. Logically,only fractures that remain propped/open over time will contributesignificantly to ultimate production and hydrocarbon recovery from thewell. Moreover, fracture geometry has importance in ultimate recovery,well spacing design, well orientation, and even in-stage (within asingle well) spacing or perforation designs and spacing. For example,stress shadowing from one fracture, perforation cluster, or fracturenetwork can reduce recovery or propensity to fracture of another nearbystage, cluster, or any adjacent well. Note that because methodsaccording to the present disclosure rely on information travelingpredominantly through the fluid and interfaces, a hydraulicallyconnected volume is where measurements may be made.

Continuously measuring pressure-related signals and also the rate ofchange of pressure (these can be pressure fluctuations, or rates ofchange in pressure such as provided by pressure gauges/transducersand/or hydrophones), how they change, their frequency characteristics,overall phase shift and time of travel, may be related to instantaneousfracture geometry. Comparing with theoretical speed of the wave giventhe proppant size (which puts a lower limit on a single fracturethickness), fracture geometry and other geophysical parameters can bedetermined.

The quality factor (Q=resonant (maximum amplitude) frequency/resonancespectral width at half maximum amplitude frequency) of resonances may beestimated and used to infer the fluid communication of fracture networksto the well.

In an embodiment according to the present disclosure, sensors are placedon the surface near, at, or contacting the fluid inside the well. Thesensors may include but are not limited to hydrophones that areconnected to the wellbore fluid when pumping, other acoustic measurementsensors (to measure ambient noises), accelerometers, pressuretransducers, jerk-meters (measure derivative of acceleration),geophones, microphones, or similar sensors. Other physical quantitiescan also be measured, such as temperature or fluid composition toprovide temperature corrections and calibrations or for data consistencychecks for all the sensors. Measuring nearby ambient surface noise usingmicrophones, geophones, accelerometers or similar sensors can help inattenuation of noise in fluid pressure or pressure time derivativesensor data (i.e. pump noise as contrasted with fluid resonances due tofractures). Sensors measuring chemical composition and density of thepumped fluid may be used to improve analysis and are thereforeimplemented in some embodiments. An example arrangement of sensors isshown in FIG. 1. Sensors may be placed on and near a well W as well asan adjacent well W1. The various sensor locations are shown at S1through S6. Sensors S1, S2, S4, S5, and S6 may be exposed to fluid beingpumped throughout a fracturing operation. A pressure or seismic source Smay be disposed at or near the position of sensor S1 and may beconnected to the well W only when necessary to activate it. Sensor(s) S3may be one or more seismic sensors disposed on the ground within about100 meter(s) of the well W, depending on available access.

Sensor(s) S1 on the wellhead may measure, e.g., pressure, pressure timederivative, temperature. Sensor(s) S2 located near fracture treatmentpumps may measure pressure, pressure time derivative, chemicalcomposition, temperature.

More than one sensor on the wellhead (e.g., at S1) is not required,however additional sensors placed proximate to the wellhead can providehigher accuracy, such as directionality of propagating signals, ambientnoise records for noise cancelling, ground vibration measurements, steelcasing vibrations, etc. and thus methods according to the presentdisclosure may benefit from using such sensors. In some embodiments allthe sensors should have substantial response at ˜1 kHz or above as wellas in the very low, even sub-Hz, frequency range.

The signals from the sensors are amplified, filtered, captured,digitized, recorded, stored, and transferred to a computer or similardevice for processing, e.g., in a recording unit R which may be disposedproximate the well W. Such recording unit R may be further connectedwith a control system CS of the entire fracturing operation to detectsensor measurements, analyze the measurements and provide possiblefeedback control loops to optimize operations and correlate multitude ofdata streams for final processing (pump rotation speeds, pumping rates,chemical input rates, blender rates, fluid density, sand concentration,etc.). This is depicted by Computing System 100, in FIG. 8.

Although data of primary interest can only be obtained in certainintervals of interest, a continuous stream of data acquired atreasonably high frequencies (up to approximately ˜100 kHz) may bebeneficial to further analysis and a continuous or near continuous, orcontinuously-pulsed measurement stream is desirable for microseismicevent rate monitoring. In particular, measurements of signals atrelatively low and subsonic frequencies (less than about 5 kHz and 20 Hzrespectively) are important for fracture characteristic analysis andprovide some of the frequency domain information. Higher frequencies mayprovide higher spatial and time resolution into the fractures and ofseismic and other subsurface events, while their penetration depth awayfrom the wellbore may not be as large. The accurate recording of lowfrequencies is also important in order to detect large fractures andlarger-scale stimulated reservoir volumes.

Such sensor attachments and connections as described may be made safelyusing common practices and design principles even though fracturingpressures are very high. Spacing of the sensors and availableconnections will be specific to a given fracturing well-configuration,but in general a sensor should be connected closer to the formation(farther from the fracturing pumps, e.g. on a wellhead). Exceptions mayinclude secondary sensor(s), e.g., S2 on the pumping flowline, that canbe correlated with the measurements made by a sensor, e.g., at S1 (S1→S2or S2→S1) to infer traveling wave linear directionality in the flowlineand thus in the well.

As stated above, more than one sensor is not required, howeveradditional sensors can provide higher accuracy, such as directionalityof propagating signals, ambient noise records for noise cancelling,ground vibrations, steel casing vibrations, etc. Thus having more thanone sensor is included in FIG. 1. Measurements from the various sensorsmay be time synchronized. One method of synchronizing sensors is usingGPS time signals at the sensors or on the recording system R (if thesensors are far apart). Combining all real-time sensor measurementstreams into a single common data acquisition unit, e.g. the recordingunit R could obtain the same objective.

Sources of signals that excite resonant frequencies in the combinedwell-fracture network will come from, including but not limited to:pumping and pumping changes; performing nearby perforations; nearbygeologic activity; AND surface or borehole-based time-limited/pulsedenergy sources. In addition, continuous sources (valves, pumps such asare already used), or micro-seismic events—microseismic or fractureactivity are broadband sources well-suited to excite such resonantfrequencies. In particular, inside reservoir induced (by ongoinghydraulic fracturing operation in the well of interest or a nearby wellwhile pumping) microseismic activity, is important in generating some ofthe signals and fracture waves.

FIG. 2 shows an example geophysical model of the well traversingsubsurface formations, fractures 24 created by or enhanced by fracturetreatment pumping, measurements obtained using a method according to thepresent disclosure and analysis of the measurements. Traveling fluidpressure waves are shown schematically at R¹ in the graph at 20 beingreflected pressure wave in the wellbore, and T representing transmittedpressure waves in the wellbore. FIG. 2 shows graphic representations ofthe transmitted pressure wave T with respect to time superimposed on thereflected pressure wave R¹ and its reverberations on the graph at 20.Frequency domain analysis is shown schematically on the graph at 22.

Measurements acquired during a fracture treatment pumping stage may besimilar in characteristics to what is shown in FIG. 3. Note that a rapidpressure change generates an acoustic signal (can be subsonic <20 Hz, orsupersonic >20 kHz) and often may be referred to as such. This signal inturn, may generate an “echo” returning from the subsurface region of thewell.

The upper frame 30 in FIG. 3 shows surface fluid pressure applied to awell with respect to time as measured, e.g., at sensor S1 in FIG. 1.Horizontal axis represents time. The middle frame 32 shows a graph ofthe time derivative of this measured pressure (hydrophone signal). Thelower frame 34 shows graphs of time derivative of the measured pressurewith reference to specific events occurring in the well and in theformations penetrated by the well. During a common hydraulic fracturingoperation, a ball-seating plug is set at a selected depth in the well,then a sealing ball is pumped down the well at a modest rate (few tensof barrels per minute, e.g., A at 100 seconds in the upper frame 30),slowing down before the ball engages a plug (e.g. B, at 195 seconds inFIG. 3). Immediately after the ball seats, at which point if chosenbased on formation composition, properly used and spotted, acid wouldreach the area of perforations in the well casing and the formation. At200 seconds in FIG. 3, the pressure rises to the point where fracturesin the newly pressured fracture treatment stage start to open. A steeppressure increase shown in the upper frame 30 indicates that the presentfracture treatment stage is hydraulically isolated from the previousfracture treatment stage.

As more fracturing fluid is pumped and the fluid pumping rate increases,fractures continue propagating in the formation. Operators typicallyincrease the rate of pumping until a target rate is reached (tens,sometimes about 100 barrels per minute-bpm), which also increases thefluid pressure. Once a target planned “sweet spot” or optimized pumpingrate is reached, the operator may maintain that pumping rate unlessunexpected behavior (pump failures, screen-out, or unexpected pressurerise) and safety considerations or feedback from methods as disclosedherein dictate otherwise. For example pressure and pumping rate can bechanged to overcome friction and to mitigate growth of fractures. Duringthis time, proppant is typically added to the pumped fluid to keepfractures open after the pressure on the fracturing fluid is relieved.Various signal profiles are shown as a way of example of signalvariation in FIG. 30D, F.

In FIG. 3, note in the lower frame 34 measurements corresponding to pumpnoise, pressure changes with ball seating, a microseismic event E,identified around 420.5 seconds and a stabilized pumping noise andpressure signals during this time. Measuring, detecting, and providingreal-time feedback of the microseismic events thus detected may also bevaluable. Coming from a single or a small number of sensors, this datacan be made readily available and processed in real-time for the benefitof the engineers performing the treatment.

In its basic form, simply knowing how many formation-breaking, i.e.microseismic, events occur per unit time may show how much the formationhas been fractured and can be combined with additional information (suchas but not limited to fully passive microseismic analysis) for even morecomprehensive understanding. Real-time aspects of the pressure andpressure time derivative measurements can be useful as the operator maywant to maintain a certain formation-breaking/fracture creating rate(microseismic events per unit time interval) to optimize fracturecreation for maximum hydrocarbon recovery.

Time-frequency analysis may be used to show change of the pressure wavespectrum over time. Frequency domain analysis, such as may be providedby a Fourier transform can then have a better resolution in thetime-frequency stationary period.

In some embodiments, measurements from a plurality of sensors such asshown in FIG. 1 comprising pressure transducers, accelerometers,hydrophones, or geophones may be used to reduce surface-based noise,reconfirm the existence of strong events, and/or to eliminate certainfrequencies in the signals such as those originating from the pumps orsurface activity instead of the reservoir and/or fractures or subsurfacesignals carried though the wellbore.

Inversion of the Measurement

The inversion of the measurement to determine physical parametersdescribing the fractures and fracture network requires a description ofhow pressure disturbance(s) interact(s) with the fractures, the fracturenetwork, the wellbore and the system comprised by these parts. Withinall elements of the system and its component parts pressure disturbancesobey a second-order in time differential equation composed of terms thatdescribe wave propagation and terms that describe diffusion behavior.The relative amplitude of each of these terms differs in the wellboreand in the fracture and fracture network.

In the wellbore, the wave propagation terms dominate and the pressuredisturbance propagates as a wave with relatively little attenuation.Except in unusual circumstances in the fracture and fracture networks,the diffusion terms dominate and the amplitude of the pressuredisturbance decays rapidly with relatively little wave-like nature. Onlyin unusual circumstances are interface waves, such as Stoneley waves,Scholte waves, Rayleigh waves, Love waves and Krauklis waves, excited,and such waves propagate within the fractures.

A specific method will now be explained to invert the data based on theabove understanding. Those skilled in the art will understand that thespecific method may be modified or extended in whole or in part. Themethod, which inverts the data based on the above understanding, willnow be explained. The explanation of an example embodiment of the methoduses a model (see, e.g., Mathieu and Toksoz, 1984; Hornby et al., 1989;Kostek et al., 1998a; Henry, 2005), to describe tube wave reflectionfrom fractures. Important elements of this disclosure refer tocomplex-valued frequency dependent reflection coefficient, proppantfilled Darcy flow, and elastic compliance of the fractures as describedin paragraphs below. Tube waves at the frequencies of interest areidealized as pressure waves obeying the wave equation with speed c_(T)(Biot, 1952). Attenuation during propagation is accounted for using afrequency-independent quality factor Q_(T), not to be confused withwellhead flow rate Q(t).

The borehole may be sealed with a packer, and fractures have beencreated through several perforation clusters in the casing. At lowfrequencies of interest, wavelengths of tube waves are sufficientlylarge that it may be assumed that all fractures effectively experiencethe same pressure at their junction with the borehole. Tube waves thusreflect from the set of fractures and packer collectively, rather thanfrom individual fractures. The tube wave reflection coefficient for thisgeometry may be determined by the expression:

$\begin{matrix}{{{R(\omega)} = \frac{{Z_{f}(\omega)} - Z_{T}}{{Z_{f}(\omega)} + Z_{T}}},} & (1)\end{matrix}$

where Z_(T)=r_(T) C_(T)/A_(T) is the tube wave hydraulic impedance (fora borehole fluid density r_(T), tube wave speed c_(T), and boreholecross-sectional area A_(T)) and Z_(f)(ω) is the hydraulic impedance ofthe set of fractures and packer that terminates the portion of theborehole that is hydraulically connected to the wellhead.

Here, R(ω) is a complex-valued, frequency-dependent reflectioncoefficient, and hydraulic impedance Z is defined as the ratio ofpressure change to change in volumetric flow rate. The wellhead pressurewith respect to time P(t), in response to an imposed wellhead flow rateQ(t), may be expressed in the frequency domain as:

$\begin{matrix}\begin{matrix}{{\hat{P}(\omega)} = {Z_{T}{\hat{Q}(\omega)}\frac{1 + {{g(\omega)}{R(\omega)}}}{1 - {{g(\omega)}{R(\omega)}}}}} \\{{= {Z_{T}{\hat{Q}(\omega)}\left\{ {1 + {2{\sum\limits_{n = 1}^{\infty}\left\lbrack {g(\omega){R(\omega)}} \right\rbrack^{n}}}} \right\}}},}\end{matrix} & (2)\end{matrix}$

for reflection coefficient R(ω) given in Eq. (1) and two-way travel timefactor g(ω) that accounts for attenuation and causality preservingdispersion (See, e.g., Aki and Richards, 2009):

$\begin{matrix}{{{g(\omega)} = {\exp \left( {{\frac{2i\; \omega \; h}{c_{T}}\left\lbrack {1 - \frac{\ln \left( {\omega/\omega_{0}} \right)}{{\pi Q}_{T}}} \right\rbrack} - \frac{{\omega }h}{c_{T}Q_{T}}} \right)}},} & (3)\end{matrix}$

where h is the borehole length and ω₀ is a reference angular frequencyat which the tube wave phase velocity equals c_(T). The second form ofEq. (2) highlights the infinite sequence of reflections. In numericaltime-domain examples to follow, we construct the solution first in thefrequency domain and then invert the transform using a fast Fouriertransform.

Single Fracture

Consider a single, one-sided fracture as a planar crack extending in thepositive x direction away from the borehole to a distance L. Thefracture has cross-sectional area A in the y-z plane (e.g., for anelliptical cross-section, A=πwH/4, with maximum width w and height H).The fluid pressure p is assumed to be uniform across this cross-section,but is permitted to vary in the x direction; i.e., p=p(x; t). Thefracture is filled with proppant (porosity φ and permeability k) andfluid (density p and dynamic viscosity μ). The volumetric flow rate offluid along the fracture in the x direction is denoted as q(x; t). Thehydraulic impedance of this fracture is defined using pressure andvolumetric flow rate at the fracture mouth, p₀(t)=p(0,t) and q₀(t)=q(0,t), respectively, as Z₀(ω)=p₀(ω)/q₀(ω).

An objective is to derive Z₀(ω) for a single, one-sided fracture.Conservation of fluid mass may be represented as:

$\begin{matrix}{{{\frac{\partial\left( {{\rho\varphi}A} \right)}{\partial t} + \frac{\partial\left( {\rho q} \right)}{\partial x}} = 0},} & (4)\end{matrix}$

assuming negligible leak-off over the short time scales of interest.Next, we rewrite (4) as an equation for pressure perturbation p(x, t)within the fracture. Perturbations are assumed sufficiently small so asto justify linearization. Following standard procedures in linearporomechanics, it may be assumed that r and f depend on the localpressure p, and define fluid and pore compressibilities asβ_(f)=ρ⁻¹(∂ρ/∂ρ) and β_(f)=ϕ⁻¹(∂ϕ/∂p) respectively.

It may also be assumed that a local elasticity relation in which changesin A depend only on the local pressure. This assumption is used inseveral simple models of hydraulic fractures (e.g., the PKN model, see,Nordgren, 1972). With this assumption, the crack compliance may bedefined as β_(A)=A⁻¹(∂A/∂p). As an example, if it is assumed that thefracture height H is much less than wavelengths characterizing thepressure perturbations in the x direction, then plane strain conditionsprevail within the plane of the cross-section. This permits use of thestandard solution for a uniformly pressurized mode I crack, for whichchanges in width Δw are related to changes in pressure Δp by Δw−(H/G*)Δpwith G*−G/(1−v) for solid shear modulus G and Poisson's ratio v. Itfollows that the crack compliance is β_(A)=(H/w)(G*)⁻¹.

The general definitions of compressibilities and the crack complianceare then used to rewrite the first term in the mass balance Eq. (4) interms of the pressurization rate ∂p/∂t. In addition, Darcy's law statesthat:

$\begin{matrix}{q = {{- \frac{kA}{\mu}}{\frac{\partial p}{\partial x}.}}} & (5)\end{matrix}$

With these substitutions, the mass balance in Eq. (4) becomes thediffusion equation for pressure perturbation p(x; t) within thefracture:

$\begin{matrix}{{{{{\rho\varphi}A\beta}\frac{\partial p}{\partial t}} = {\frac{\partial}{\partial x}\left( {\frac{\rho {kA}}{\mu}\frac{\partial\rho}{\partial x}} \right)}},} & (6)\end{matrix}$

where β−β_(f)+β_(f)+β_(A) is the total compressibility/compliance. Thediffusivity and diffusion length are, respectively:

$\begin{matrix}{D = {{\frac{k}{\mu\varphi\beta}\mspace{14mu} {and}\mspace{14mu} L_{D}} = {\sqrt{D/\omega}.}}} & (7)\end{matrix}$

Consistent with the assumption of small perturbations, Eq. (6) islinearized and all coefficients (i.e., S, p, k, A and μ) are evaluatedat reference conditions. In all examples below, one may assume spatiallyuniform properties.

When the fracture is much longer than the diffusion length (L_(D)<<L),as is typically the case in our experience, the solution to Eq. (6) forimposed volumetric flow rate q₀(t) at the fracture mouth x=0 is, in thefrequency domain:

$\begin{matrix}{{\hat{p}\left( {x,\omega} \right)} = {{{\hat{q}}_{0}(\omega)}\frac{\mu}{kA}\sqrt{\frac{D}{{- i}\; \omega}}{{\exp \left( {{- \sqrt{\frac{{- i}\; \omega}{D}}}x} \right)}.}}} & (8)\end{matrix}$

The hydraulic impedance of this single, one-sided fracture is:

$\begin{matrix}{{Z_{0}(\omega)} = {{\frac{\mu}{kA}\sqrt{\frac{D}{{- i}\; \omega}}} = {\sqrt{\frac{\mu}{{- i}\; \omega \; \varphi \; \beta \; {kA}^{2}}}.}}} & (9)\end{matrix}$

Multiple Fractures

Now consider a small section of the borehole hydraulically connected toa set of N fractures, each extending bilaterally away from the borehole,and terminated by an impermeable, rigid plug. Elastic interactionsbetween the fractures are neglected. It may be assumed that allfractures experience the same pressure p₀(t) at their junction with theborehole, and one may define q_(i)(t) as the volumetric flow rate intofracture i (i=1, . . . , N). The hydraulic impedance of the fracture setis:

$\begin{matrix}{{{Z_{f}(\omega)} = \frac{{\hat{p}}_{0}(\omega)}{2{\sum\limits_{i = 1}^{N}{{\hat{q}}_{i}(\omega)}}}},} & (10)\end{matrix}$

where the denominator in Eq. (10) is the total volumetric flow rate intoall N fractures, and the factor of two is because the fractures extendlaterally from both sides of the borehole (x>0 and x<0). If it isfurther assumed that all fractures are identical, each having hydraulicimpedance Z₀(ω), then Z_(f)(ω)=Z₀(ω)/2N.

Compliant, Elliptical Crack Model

As a specific example, suppose that the compressibility or compliance bis dominated by the crack compliance β_(A), such that β˜(H/w)(G*)⁻¹.Using this expression, and assuming elliptical cross-section (A=πwH/4),the hydraulic impedance of N bilateral fractures, in the small diffusionlength limit of Eq. (9), reduces to

$\begin{matrix}{{Z_{f}(\omega)} = {\frac{2}{\pi N}{\sqrt{\frac{G*{\mu\varphi}}{{- i}\; \omega \; k\; {wH}^{3}}}.}}} & (11)\end{matrix}$

Eq. (11) will be used in the remainder of this disclosure, together withEq. (1) and Eq. (2), to interpret data.

Active Source Measurement

FIG. 4A shows a representative active source hydrophone time seriesalong with the best-fitting model. The source is idealized as a Gaussianmodulation of wellhead flow rate, Q(t)˜exp(−(ωT)²/2), for sourceduration T. Setting c_(T)=1460 m/s, h=4805 m, G=13:3 GPa, N=6, H=10 m,ϕ=0.5, and μ=5×10⁻³ Pa s, one may then vary the fracture conductivitykw, borehole quality factor Q_(T), source duration T, and sourceamplitude to minimize the waveform misfit in the L₂ norm. It may bedetermined that Q_(T)˜70, T˜0.055 s, and kw˜0.38 D m.

FIG. 4B illustrates how conductivity kw affects waveforms. This isbecause the reflection coefficient R depends on kw as shown in FIG. 4C.The real part of R is negative, and R→1 at high frequencies and forhighly conductive fractures (large kw). In this limit, the fracturehydraulic impedance is much less than the tube wave impedance(Z_(f)<<Z_(T)), such that waves reflect as if from a constant pressure(i.e., “open”) end. At lower frequencies, and also for less conductivefractures (smaller kw), the fracture hydraulic impedance increases, andthe reflection coefficient shows appreciable differences from theopen-end limit. For even smaller kw than shown in the FIG. 4C,Z_(f)>>Z_(T) and R→1 (i.e., “closed” end).

The inferred value for conductivity, kw˜0.38 D m, is reasonablyconsistent with independent estimates of width w and proppant packpermeability k. First, it should be emphasized that the measurementalone cannot provide separate constraints on k and w. For example, theinferred conductivity is consistent with w=1 mm and k=400 D, w=1 cm andk=40 D, or w=0.1 m and k=4 D. Laboratory measurements of proppant packpermeability (See, e.g., Lee et al., 2010) show values around 100 D, forwhich the inferred width is 4 mm.

Water Hammer Measurement

Next, t data may be interpreted in the frequency domain. FIG. 5A showsthe hydrophone Fourier spectrum from water hammer produced when pumpsare shut off at the end of the stage (ISIP water hammer). The multiplespectral peaks are the resonant modes of the borehole-fracture system.The resonance frequencies of open- and close-ended tubes are well known.The present example embodiment of a model predicts a continuoustransition between these limits as the hydraulic impedance ratio,Z_(f)/Z_(T), is varied. FIG. 5B shows the sensitivity of modeled spectrato fracture conductivity kw.

To demonstrate this, one may apply the model with the same parameters asbefore but with the source flow rate Q(t) idealized as a step function.FIG. 5B shows graphically how kw influences the spectra. The resonancefrequencies transition from the closed-end limit for small kw to theopen-end limit for large kw. Since the actual source time function ismore complicated than a step function, the model may be fit to the databy matching the frequencies and quality factors of individualresonances, rather than attempting to directly match the spectrum. Thisprocedure, not illustrated here, provides values reasonably consistentwith those inferred from the active source full waveform inversion.

Interpretation of the Inversion Results

In the preceding section is described one specific method of invertingthe data for a parameter kw/μ which controls the rate at which fluidflows into and out of the fracture and which may be designated as theconductivity of the fracture or fracture network. This is a relevantfactor in the subsequent production of hydrocarbons.

In addition, by repeating this measurements at least two distinct timesbefore, during or after the pumping of a fracture treatment, it ispossible to calculate the change, or rate of change, of the conductivitywhich provides information on the effectiveness of the fracturingtreatment. The initial, “baseline” measurement may also be taken fromanother dataset of a well having similar well_parameters and formationsto estimate such a change.

In addition, by examining the conductivity calculated from resonances atcomparatively low frequencies, intermediate frequencies and highfrequencies can be analyzed. Different frequencies are sensitive todifferent ranges of investigation with low frequencies extendingfurthest and high frequencies extending the least distances. Thus, fromcomparison of the conductivity estimates made at different frequenciesit is possible to estimate the conductivity, conductivity changes andrates of conductivity change at different distances from theperforations and wellbore.

Thus, the calculation of conductivities and their change with respect totime can be interpreted as originating from the spatial distribution ofchanges in conductivity, and consequently one can infer the distributionof proppant and its change with time.

Furthermore, the distribution of proppant as a function of distance fromthe perforation can be interpreted in terms of the complexity of thefracture network. A situation where the known total volume of proppantis distributed equally with respect to distance from the well isexpected to be the result of a relatively simple fracture network.Conversely, when the known total volume of proppant is highlyconcentrated near the well it is expected that a complex fracturenetwork exists. This complex (“tortuous”) fracture network provides boththe volume to contain the proppant and the complexity which traps theproppant and prevents it from being carried further from the well.

Furthermore, it is possible to identify segments of a borehole (stages)that contain fractures that exhibit significantly larger, orsignificantly smaller changes in conductivity caused by hydraulicfracturing. These segments can be correlated, or otherwise associated,with particular geological characteristics of the formation in which theborehole is situated. These geological characteristics are typicallydetermined from lithological logs, or other logs (e.g.,rate-of-penetration logs) recorded while drilling the borehole or usingdata acquired after drilling acquired on, for example, wireline. Oncethis correlation, or association, of fractures yielding high, or low,conductivities with particular features of lithological or other logshas been established, it can then be used to plan perforation andhydraulic fracture location in other boreholes to optimize operations.For example, if it is established by correlation or other comparisonthat portions of the well that exhibited low rates-of-penetration whiledrilling also tend to produce high conductivity fractures, then insubsequent wells it may be possible to locate the perforations (wherethe fractures originate) in segments of the well that exhibited lowrates-of-penetration when they were drilled. As another example, if itis established that high conductivity fractures are associated withsilica-rich portions of the formation, in future wells one can positionthe perforations primarily in silica-rich segments of the well; thenhigh conductivity fractures may be expected in other wells drilledthrough silica-rich formations or portions thereof. Many othercorrelations between fracture characteristics and lithology (formationmineral composition) or geomechanical characteristics (e.g., bulk andelastic moduli, Poisson's ratio, compressive and tensile strength) ofthe formation are possible.

FIG. 6 applies the method disclosed above to a completed pumped fracturetreatment stage where the well is closed to flow (“shut in”) aftercompleting fracture treatment. Conductivity kw may be determined makingsome reasonable assumptions about certain material properties, such asshear moduli and Poisson's ratios of the fractures formation. Such kwdetermination may not be exact, but if consistent values are used foreach fracture stage stage in a multi-stage fracture treatment, thecomparison will be representative of comparable fracture conductivity,kw, across multiple fracture treatment stages. The fracture conductivityas may be observed in FIG. 6 tends to decrease over time as fracturesslowly close and the formation absorbs some of the pumped fracturefluid. Various fracture closure pressures can be estimated based on alevel of conductivity reached. An ultimate or terminal value of kw canbe projected as an asymptotic value of the tail of the determinedconductivity curve, e.g., the curve in FIG. 6, projected to a relativelylong time (several days to several weeks) after the end of pumping ofthe fracture treatment.

FIG. 7 applies the kw determinations and conductivity with respect totime for two completed fracture treatment stages containing gels as aprincipal component of the pumped fracturing fluid. Gels for fracturetreatment are relatively viscous (pseudo-solid) to carry high proppantconcentration so as to place the proppant in the induced fractures, butare designed to decompose or “break down” after a period of time beingpresent in fractures in a subsurface formation. The graph in FIG. 7shows how the time when the gel in the formation breaks down can bedetermined. Gel breakdown may be identified as occurring at the timewhen fracture conductivity increases markedly after being substantiallyconstant for a determinable length of time.

Although long-term post shut-in (end of fracture treatment and holdingof the final pumped pressure in the well) monitoring of parameters maynot always be possible in wells in which fracture treatments are to beperformed sequentially in multiple stages without considerable delaybetween successively pumped stages, the final fracture treatment stageand (if available any other available stage(s)) can be measured and kwvalues may extrapolated for the remainder of the well. Such kw valuescan aid completion engineers to design more suitable gel mixes or inputfracture and leak off measurements to better inform fracture models.

Moreover, as fractures close and pumped fluid leaks out while theproppant takes on the pressure of the formation, there will be a steadystate reached after some time (similar to what is seen in FIG. 6 at 300minutes). This may represent a final state of the well with onlyfractures having proppant properly placed remaining open. Such ameasurement of conductivity can then be either extrapolated or directlymeasured for a stage, well, or for a formation in general.

The methods illustrated with reference to FIGS. 6 and 7 may be used todetermine a start time for “flowback and cleanup” operations (i.e.,initiating flow from the formation through the well) and subsequentproduction of fluid from the formation, i.e., commencing production fromthe well.

Although the methods shown graphically in FIGS. 6 and 7 are illustratedas being performed visually using graphs of fracture conductivity withrespect to time, it will be appreciated by those skilled in the art thatequivalent methods may be performed by a suitably programmed computer orcomputer system and the relevant times and kw values may be presented inany form, including, for example and without limitation, as numericaltables, identified times and values associated with identified events.

Measurements as described in this disclosure can be taken at anyarbitrary time after fracturing concludes. Later on, when at least onestage, or entire well is brought on production, these same measurementscan be repeated to monitor conductivity of the well over time period ofweeks or months, as the well produces fluid. Even though in this case,production from several or all stages and conductivities from all stageswould be co-mingled, this data can be used to monitor and correlate wellproduction over time. In addition, the sensors may also passively listento signals coming naturally from the formation as the well is beingproduced. Those signals (or noise levels) can be used in additionalanalysis, for example using the computer system described in FIG. 8, tomonitor production.

FIG. 8 shows an example computing system 100 in accordance with someembodiments. The computing system 100 may be an individual computersystem 101A or an arrangement of distributed computer systems. Theindividual computer system 101A may include one or more analysis modules102 that may be configured to perform various tasks according to someembodiments, such as the tasks explained with reference to FIGS. 2, 3,4A, 4B, 4C, 5A, 5B, 6 and 7. To perform these various tasks, theanalysis module 102 may operate independently or in coordination withone or more processors 104, which may be connected to one or morestorage media 106. A display device 105 such as a graphic user interfaceof any known type may be in signal communication with the processor 104to enable user entry of commands and/or data and to display results ofexecution of a set of instructions according to the present disclosure.

The processor(s) 104 may also be connected to a network interface 108 toallow the individual computer system 101A to communicate over a datanetwork 110 with one or more additional individual computer systemsand/or computing systems, such as 101B, 101C, and/or 101D (note thatcomputer systems 101B, 101C and/or 101D may or may not share the samearchitecture as computer system 101A, and may be located in differentphysical locations, for example, computer systems 101A and 101B may beat a well drilling location, while in communication with one or morecomputer systems such as 101C and/or 101D that may be located in one ormore data centers on shore, aboard ships, and/or located in varyingcountries on different continents).

A processor may include, without limitation, a microprocessor,microcontroller, processor module or subsystem, programmable integratedcircuit, programmable gate array, or another control or computingdevice.

The storage media 106 may be implemented as one or morecomputer-readable or machine-readable storage media. Note that while inthe example embodiment of FIG. 6 the storage media 106 are shown asbeing disposed within the individual computer system 101A, in someembodiments, the storage media 106 may be distributed within and/oracross multiple internal and/or external enclosures of the individualcomputing system 101A and/or additional computing systems, e.g., 101B,101C, 101D. Storage media 106 may include, without limitation, one ormore different forms of memory including semiconductor memory devicessuch as dynamic or static random access memories (DRAMs or SRAMs),erasable and programmable read-only memories (EPROMs), electricallyerasable and programmable read-only memories (EEPROMs) and flashmemories; magnetic disks such as fixed, floppy and removable disks;other magnetic media including tape; optical media such as compact disks(CDs) or digital video disks (DVDs); or other types of storage devices.Note that computer instructions to cause any individual computer systemor a computing system to perform the tasks described above may beprovided on one computer-readable or machine-readable storage medium, ormay be provided on multiple computer-readable or machine-readablestorage media distributed in a multiple component computing systemhaving one or more nodes. Such computer-readable or machine-readablestorage medium or media may be considered to be part of an article (orarticle of manufacture). An article or article of manufacture can referto any manufactured single component or multiple components. The storagemedium or media can be located either in the machine running themachine-readable instructions, or located at a remote site from whichmachine-readable instructions can be downloaded over a network forexecution.

It should be appreciated that computing system 100 is only one exampleof a computing system, and that any other embodiment of a computingsystem may have more or fewer components than shown, may combineadditional components not shown in the example embodiment of FIG. 8,and/or the computing system 100 may have a different configuration orarrangement of the components shown in FIG. 8. The various componentsshown in FIG. 8 may be implemented in hardware, software, or acombination of both hardware and software, including one or more signalprocessing and/or application specific integrated circuits.

Further, the acts of the processing methods described above may beimplemented by running one or more functional modules in informationprocessing apparatus such as general purpose processors or applicationspecific chips, such as ASICs, FPGAs, PLDs, or other appropriatedevices. These modules, combinations of these modules, and/or theircombination with general hardware are all included within the scope ofthe present disclosure.

REFERENCES CITED IN THIS DISCLOSURE

Aki, K., and P. G. Richards, 2009, Quantitative Seismology: UniversityScience Books.

Biot, M., 1952, Propagation of elastic waves in a cylindrical borecontaining a fluid: Journal of Applied Physics, 23, 997-1005.

Henry, F., 2005, Characterization of borehole fractures by the body andinterface waves: TU Delft, Delft University of Technology.

Hornby, B., D. Johnson, K. Winkler, and R. Plumb, 1989, Fractureevaluation using reflected Stoneley-wave arrivals: Geophysics,54,1274-1288.

Kostek, S., D. L. Johnson, and C. J. Randall, 1998a, The interaction oftube waves with borehole fractures, part i: Numerical models:Geophysics,63, 800-808.

Lee, D. S., D. Elsworth, H. Yasuhara, J. D. Weaver, and R. Rickman,2010, Experiment and modeling to evaluate the effects of proppant-packdiagnosis on fracture treatments: Journal of Petroleum Science andEngineering,74, 67-76.

Mathieu, F., and M. Toksoz, 1984, Application of full waveform acousticlogging data to the estimation of reservoir permeability: Technicalreport, Massachusetts Institute of Technology. Earth ResourcesLaboratory.

Nordgren, R., 1972, Propagation of a vertical hydraulic fracture:Society of Petroleum Engineers,12, 306-314.

Although only a few examples have been described in detail above, thoseskilled in the art will readily appreciate that many modifications arepossible in the examples. Accordingly, all such modifications areintended to be included within the scope of this disclosure as definedin the following claims.

What is claimed is:
 1. A method for characterizing a hydraulic fractureor network of fractures in a subsurface formation, comprising: inducinga pressure change in a well drilled through the subsurface formation,the pressure change inducing tube waves in the well; measuring at alocation proximate to a wellhead at least one of pressure and a timederivative of pressure in the well for a selected length of time;determining a conductivity of at least one fracture, using the measuredat least one of pressure and the time derivative of pressure; anddetermining a change in the determined conductivity with respect totime.
 2. The method of claim 1 wherein the inducing a pressure changecomprises pumping a hydraulic fracture treatment.
 3. The method of claim1 wherein the inducing a pressure change comprises inducing water hammerby changing a flow rate of fluid into or out of the well.
 4. The methodof claim 1 wherein the inducing a pressure change comprises operating anacoustic source which propagates a pressure pulse into fluid within thewell.
 5. The method of claim 1 further comprising determining a fractureclosure rate using the determined change in fracture conductivity withrespect to time.
 6. The method of claim 5 wherein the fracture closurepressures are determined by aligning pressure measurement withconductivity decrease.
 7. The method of claim 5 determining fluid leakoff rates by correlating with the conductivity decrease over time. 8.The method of claim 1 further comprising initiating flowback, cleanupand production from the well after the change in conductivity isdetermined.
 9. The method of claim 1 further comprising determining agel breakdown time from a determined time of substantial increase infracture conductivity.
 10. The method of claim 9 further comprisinginitiating flowback, cleanup and production from the well after the gelbreakdown time is determined.
 11. A method for characterizing ahydraulic fracture or network of fractures in a subsurface formation,comprising: inducing a pressure change in a well drilled through thesubsurface formation, the pressure change inducing tube waves in thewell; measuring at a location proximate to a wellhead at least one ofpressure and a time derivative of pressure in the well for a selectedlength of time; in a computer, determining a conductivity of at leastone fracture, using the measured at least one of pressure and the timederivative of pressure; and in the computer, determining a change in thedetermined conductivity with respect to time.
 12. The method of claim 11wherein the inducing a pressure change comprises pumping a hydraulicfracture treatment.
 13. The method of claim 11 wherein the inducing apressure change comprises inducing water hammer by changing a flow rateof fluid into or out of the well.
 14. The method of claim 11 wherein theinducing a pressure change comprises operating an acoustic source whichpropagates a pressure pulse into fluid within the well.
 15. The methodof claim 11 further comprising, in the computer, determining a fractureclosure rate using the determined change in fracture conductivity withrespect to time.
 16. The method of claim 15 wherein the fracture closurepressures are determined in the computer by correlating pressuremeasurement with a conductivity change.
 17. The method of claim 15further comprising, determining in the computer fluid leak off rates bycorrelating with the conductivity decrease over time.
 18. The method ofclaim 11 further comprising initiating flowback, cleanup and productionfrom the well after the change in conductivity is determined.
 19. Themethod of claim 11 further comprising, in the computer, determining agel breakdown time from a determined time of substantial increase infracture conductivity.
 20. The method of claim 19 further comprisinginitiating flowback, cleanup and production from the well after the gelbreakdown time is determined.
 21. The method of claim 11 wheremeasurements are made during flowback and production.